Definition 111.24.1. Let $\phi : A \to B$ be a homomorphism of rings. We say that the going-up theorem holds for $\phi$ if the following condition is satisfied:

• for any ${\mathfrak p}, {\mathfrak p}' \in \mathop{\mathrm{Spec}}(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P \in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}$, there exists $P'\in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}'$ such that $P \subset P'$.

Similarly, we say that the going-down theorem holds for $\phi$ if the following condition is satisfied:

• for any ${\mathfrak p}, {\mathfrak p}' \in \mathop{\mathrm{Spec}}(A)$ such that ${\mathfrak p} \subset {\mathfrak p}'$, and for any $P' \in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}'$, there exists $P\in \mathop{\mathrm{Spec}}(B)$ lying over ${\mathfrak p}$ such that $P \subset P'$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).